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  得到APP网页版
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  echarts示例
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  AJ-Report
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  码云Git大全
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  得到APP网页版
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  <span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></li></ul></div></div> <!----></nav>  <ul class="sidebar-links"><li><a href="/general/git.html" class="sidebar-link">git 使用攻略</a></li><li><a href="/general/markdown.html" class="sidebar-link">Markdown基本语法</a></li><li><a href="/general/nexus.html" class="sidebar-link">nexus3搭建私有源</a></li><li><a href="/general/latex.html" aria-current="page" class="active sidebar-link">Katex 公式格式速查</a><ul class="sidebar-sub-headers"><li class="sidebar-sub-header"><a href="/general/latex.html#公式格式" class="sidebar-link">公式格式</a></li><li class="sidebar-sub-header"><a href="/general/latex.html#四则运算" class="sidebar-link">四则运算</a></li><li class="sidebar-sub-header"><a href="/general/latex.html#指数对数" class="sidebar-link">指数对数</a></li><li class="sidebar-sub-header"><a href="/general/latex.html#表达式" class="sidebar-link">表达式</a></li><li class="sidebar-sub-header"><a href="/general/latex.html#计算式" class="sidebar-link">计算式</a></li><li class="sidebar-sub-header"><a href="/general/latex.html#累加累积" class="sidebar-link">累加累积</a></li><li class="sidebar-sub-header"><a href="/general/latex.html#积分、微分与极限" class="sidebar-link">积分、微分与极限</a></li><li class="sidebar-sub-header"><a href="/general/latex.html#强调与分隔" class="sidebar-link">强调与分隔</a></li><li class="sidebar-sub-header"><a href="/general/latex.html#二元运算符" class="sidebar-link">二元运算符</a></li><li class="sidebar-sub-header"><a href="/general/latex.html#希腊字母" class="sidebar-link">希腊字母</a></li><li class="sidebar-sub-header"><a href="/general/latex.html#latex常用大型公式模板" class="sidebar-link">Latex常用大型公式模板</a></li></ul></li><li><a href="/general/vscode.html" class="sidebar-link">VS Code 常用操作</a></li><li><a href="/general/guide.html" class="sidebar-link">搭建指南</a></li><li><a href="/general/reg.html" class="sidebar-link">正则表达式</a></li></ul> </aside> <main class="page"> <div class="theme-default-content content__default"><h1 id="katex-公式格式速查"><a href="#katex-公式格式速查" class="header-anchor">#</a> Katex 公式格式速查</h1> <blockquote><p><a href="https://katex.org/" target="_blank" rel="noopener noreferrer">KaTeX<span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a>是目前<em>最快</em>的网络数学排版库, KaTeX 的布局是基于 Donald Knuth 的 TeX，这是数学排版的黄金标准。</p></blockquote> <p>Katex 数学公式语法的完全版见<a href="https://blog.csdn.net/Leytton/article/details/103745169/" target="_blank" rel="noopener noreferrer">KaTeX 数学公式语法<span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a>, 本文仅列出使用频率比较高的部分.</p> <h2 id="公式格式"><a href="#公式格式" class="header-anchor">#</a> 公式格式</h2> <ul><li>行内公式排版, 通常用单个<code>$</code>符号包裹, 渲染后与正文处于同一段落, 如下所示:</li></ul> <div class="language-tex extra-class"><pre class="language-tex"><code><span class="token equation string">$c=a'+b'$</span>
</code></pre></div><p>渲染为: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi><mo>=</mo><msup><mi>a</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>+</mo><msup><mi>b</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup></mrow><annotation encoding="application/x-tex">c=a'+b'</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:0.835222em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathit">c</span><span class="mrel">=</span><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathit">b</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></p> <ul><li>块公式排版, 通常用两个<code>$$</code>符号包裹, 渲染后单独成为一段, 并居中, 如下所示:</li></ul> <div class="language-tex extra-class"><pre class="language-tex"><code><span class="token equation string">$$z = x^2 + y^2$$</span>
</code></pre></div><p>渲染为:</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>z</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">z = x^2 + y^2
</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8641079999999999em;"></span><span class="strut bottom" style="height:1.0585479999999998em;vertical-align:-0.19444em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord mathit" style="margin-right:0.04398em;">z</span><span class="mrel">=</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span></p> <p>加粗字母一般使用<code>\mathbf{}</code>包裹</p> <div class="language-tex extra-class"><pre class="language-tex"><code><span class="token equation string">$<span class="token equation-command regex">\mathbf</span>{JKLMNOPQR <span class="token equation-command regex">\Delta</span>}$</span>
</code></pre></div><p>渲染为：<br> <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mi mathvariant="bold">J</mi><mi mathvariant="bold">K</mi><mi mathvariant="bold">L</mi><mi mathvariant="bold">M</mi><mi mathvariant="bold">N</mi><mi mathvariant="bold">O</mi><mi mathvariant="bold">P</mi><mi mathvariant="bold">Q</mi><mi mathvariant="bold">R</mi><mi mathvariant="bold">Δ</mi></mrow></mrow><annotation encoding="application/x-tex">\mathbf{JKLMNOPQR \Delta}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68611em;"></span><span class="strut bottom" style="height:0.8805499999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord mathbf">J</span><span class="mord mathbf">K</span><span class="mord mathbf">L</span><span class="mord mathbf">M</span><span class="mord mathbf">N</span><span class="mord mathbf">O</span><span class="mord mathbf">P</span><span class="mord mathbf">Q</span><span class="mord mathbf">R</span><span class="mord mathbf">Δ</span></span></span></span></span></p> <div class="custom-block warning"><p class="custom-block-title">注意</p> <p>块公式必须要单起一行进行书写, 否则无法解析</p></div> <h2 id="四则运算"><a href="#四则运算" class="header-anchor">#</a> 四则运算</h2> <ul><li>加法<code>+</code>, 减法<code>-</code>, 点乘<code>\cdot</code>, 例如<code>x_2\cdot3y</code>渲染为: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>⋅</mo><mn>3</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">x_2\cdot3y</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">⋅</span><span class="mord mathrm">3</span><span class="mord mathit" style="margin-right:0.03588em;">y</span></span></span></span>, 叉乘<code>\times</code>, 例如<code>2\times3</code>渲染为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn><mo>×</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">2\times3</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">2</span><span class="mbin">×</span><span class="mord mathrm">3</span></span></span></span>.</li> <li>分子分母线<code>\frac{分子}{分母}</code>, 例如<code>\frac{a}{b}</code>渲染为: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{a}{b}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.695392em;"></span><span class="strut bottom" style="height:1.040392em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">b</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">a</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>.</li></ul> <h2 id="指数对数"><a href="#指数对数" class="header-anchor">#</a> 指数对数</h2> <ul><li>指数<code>{}^{}</code>, 第一个大括号内为底数，第二个大括号内为指数, 例如<code>3^2</code>渲染为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mn>3</mn><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">3^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathrm">3</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></li> <li>对数<code>\log_{}</code>, 大括号内为真数, 例如<code>\log_{3}{4}</code>渲染为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>log</mi><mrow><mn>3</mn></mrow></msub><mrow><mn>4</mn></mrow></mrow><annotation encoding="application/x-tex">\log_{3}{4}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.9388799999999999em;vertical-align:-0.24444em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="vlist"><span style="top:0.24444em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord textstyle uncramped"><span class="mord mathrm">4</span></span></span></span></span></li> <li>根号\sqrt{}, 例如<code>\sqrt{x^2}</code>渲染为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{x^2}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.9550540000000001em;"></span><span class="strut bottom" style="height:1.04em;vertical-align:-0.08494600000000002em;"></span><span class="base textstyle uncramped"><span class="sqrt mord"><span class="sqrt-sign" style="top:-0.11505399999999999em;"><span class="style-wrap reset-textstyle textstyle uncramped">√</span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.289em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:-0.875054em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span></span></span></span>, 高次根号<code>\sqrt[]{}</code>, 中括号内为次数, 例如<code>\sqrt[4]{a}</code>渲染为: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mroot><mrow><mi>a</mi></mrow><mrow><mn>4</mn></mrow></mroot></mrow><annotation encoding="application/x-tex">\sqrt[4]{a}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8002800000000001em;"></span><span class="strut bottom" style="height:1.04em;vertical-align:-0.23972em;"></span><span class="base textstyle uncramped"><span class="sqrt mord"><span class="root"><span class="vlist"><span style="top:-0.3363360000000001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord mathrm">4</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sqrt-sign" style="top:0.03971999999999998em;"><span class="style-wrap reset-textstyle textstyle uncramped">√</span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mord mathit">a</span></span></span><span style="top:-0.72028em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span></span></span></span></li></ul> <h2 id="表达式"><a href="#表达式" class="header-anchor">#</a> 表达式</h2> <p>将整体作为上标或者下标, 在公式需要将整体用大括号<code>{}</code>包裹:</p> <ul><li><code>$x_1$</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></li> <li><code>$x^2$</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">x^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></li> <li><code>$y^{2x}$</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>y</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">y^{2x}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.008548em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">2</span><span class="mord mathit">x</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></li> <li><code>$a_1^2$</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>a</mi><mn>1</mn><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">a_1^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.0622159999999998em;vertical-align:-0.24810799999999997em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.24810799999999997em;margin-left:0em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></li></ul> <h2 id="计算式"><a href="#计算式" class="header-anchor">#</a> 计算式</h2> <ul><li>根号<code>\sqrt{n}</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{n}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8002800000000001em;"></span><span class="strut bottom" style="height:1.04em;vertical-align:-0.23972em;"></span><span class="base textstyle uncramped"><span class="sqrt mord"><span class="sqrt-sign" style="top:0.03971999999999998em;"><span class="style-wrap reset-textstyle textstyle uncramped">√</span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mord mathit">n</span></span></span><span style="top:-0.72028em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span></span></span></span></li> <li>大(小)于号<code>\geq \leq</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≥</mo><mo>≤</mo></mrow><annotation encoding="application/x-tex">\geq  \leq</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.63597em;"></span><span class="strut bottom" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≥</span><span class="mrel">≤</span></span></span></span></li></ul> <h2 id="累加累积"><a href="#累加累积" class="header-anchor">#</a> 累加累积</h2> <ul><li>累加使用<code>\sum_{下标}^{上标}</code>, 如果上标字母是单字母, 则可以省略大括号, <code>x_i=\sum_{j=1}^ma_{ij}</code>渲染为: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">x_i=\sum_{j=1}^ma_{ij}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1.186118em;vertical-align:-0.436118em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mop"><span class="op-symbol small-op mop" style="top:-0.0000050000000000050004em;">∑</span><span class="vlist"><span style="top:0.30001em;margin-left:0em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.05724em;">j</span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.364em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">m</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></li> <li>累积使用<code>\prod_{下标}^{上标}</code>, 例如<code>\prod_{a}^{b}</code>渲染为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∏</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\prod_{a}^{b}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8501079999999999em;"></span><span class="strut bottom" style="height:1.150118em;vertical-align:-0.30001em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="op-symbol small-op mop" style="top:-0.0000050000000000050004em;">∏</span><span class="vlist"><span style="top:0.30001em;margin-left:0em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">a</span></span></span></span><span style="top:-0.364em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">b</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></li></ul> <h2 id="积分、微分与极限"><a href="#积分、微分与极限" class="header-anchor">#</a> 积分、微分与极限</h2> <ul><li>单重积分<code>\int_{a}^{b}</code>渲染为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∫</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\int_{a}^{b}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.905108em;"></span><span class="strut bottom" style="height:1.261228em;vertical-align:-0.35612em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.35612em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">a</span></span></span></span><span style="top:-0.41900000000000004em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">b</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>, 双重积分<code>\iint_{D}</code>渲染为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mo>∬</mo><mrow><mi>D</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\iint_{D}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.805em;"></span><span class="strut bottom" style="height:1.161em;vertical-align:-0.356em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0004999999999999727em;">∬</span><span class="vlist"><span style="top:0.356em;margin-right:0.05em;margin-left:-0.19445em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></li> <li>极限<code>\lim_{a \to b}</code>渲染为: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>lim</mi><mrow><mi>a</mi><mo>→</mo><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\lim_{a \to b}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="mop">lim</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">a</span><span class="mrel">→</span><span class="mord mathit">b</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></li></ul> <h2 id="强调与分隔"><a href="#强调与分隔" class="header-anchor">#</a> 强调与分隔</h2> <ul><li>a 一撇<code>a'</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>a</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup></mrow><annotation encoding="application/x-tex">a'</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:0.751892em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>, a 两撇<code>a''</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>a</mi><mrow><mi mathvariant="normal">′</mi><mi mathvariant="normal">′</mi></mrow></msup></mrow><annotation encoding="application/x-tex">a''</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:0.751892em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></li> <li>括号<code>[]</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>[</mo><mo>]</mo></mrow><annotation encoding="application/x-tex">[]</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">[</span><span class="mclose">]</span></span></span></span>, <code>()</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mo>)</mo></mrow><annotation encoding="application/x-tex">()</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mclose">)</span></span></span></span>, <code>\{\}</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>{</mo><mo>}</mo></mrow><annotation encoding="application/x-tex">\{\}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">{</span><span class="mclose">}</span></span></span></span></li> <li>括号也可以用<code>\left</code>和<code>\right</code>来增强, 例如<code>\left( \right)</code>, 这种模式下括号大小会随内容变化, 例如<code>\left(\frac{1}{2}\right)</code>渲染为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mo fence="true">(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(\frac{1}{2}\right)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.85em;"></span><span class="strut bottom" style="height:1.20001em;vertical-align:-0.35001em;"></span><span class="base textstyle uncramped"><span class="minner textstyle uncramped"><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span></li></ul> <h2 id="二元运算符"><a href="#二元运算符" class="header-anchor">#</a> 二元运算符</h2> <ul><li><code>\in</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.5391em;"></span><span class="strut bottom" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mrel">∈</span></span></span></span>, <code>\ni</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∋</mo></mrow><annotation encoding="application/x-tex">\ni</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.5391em;"></span><span class="strut bottom" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mrel">∋</span></span></span></span>, <code>\notin</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∉</mo></mrow><annotation encoding="application/x-tex">\notin</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0em;"></span><span class="strut bottom" style="height:0em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mrel">∉</span></span></span></span></li> <li><code>\subset</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.5391em;"></span><span class="strut bottom" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mrel">⊂</span></span></span></span>, <code>\subseteq</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>⊆</mo></mrow><annotation encoding="application/x-tex">\subseteq</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.63597em;"></span><span class="strut bottom" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">⊆</span></span></span></span>, <code>\supset</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>⊃</mo></mrow><annotation encoding="application/x-tex">\supset</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.5391em;"></span><span class="strut bottom" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mrel">⊃</span></span></span></span>, <code>\supseteq</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>⊇</mo></mrow><annotation encoding="application/x-tex">\supseteq</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.63597em;"></span><span class="strut bottom" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">⊇</span></span></span></span></li> <li><code>\le</code>:<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\le</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.63597em;"></span><span class="strut bottom" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≤</span></span></span></span>, <code>\ge</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≥</mo></mrow><annotation encoding="application/x-tex">\ge</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.63597em;"></span><span class="strut bottom" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≥</span></span></span></span></li></ul> <h2 id="希腊字母"><a href="#希腊字母" class="header-anchor">#</a> 希腊字母</h2> <ul><li><code>\alpha</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.0037em;">α</span></span></span></span>, <code>\beta</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.05278em;">β</span></span></span></span>, <code>\gamma</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.05556em;">γ</span></span></span></span></li> <li><code>\delta</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03785em;">δ</span></span></span></span>, <code>\epsilon</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">ϵ</span></span></span></span>, <code>\eta</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">η</span></span></span></span></li> <li><code>\theta</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">θ</span></span></span></span>, <code>\mu</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit">μ</span></span></span></span>, <code>\pi</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">π</span></span></span></span></li> <li><code>\sigma</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">σ</span></span></span></span>, <code>\tau</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.1132em;">τ</span></span></span></span>, <code>\omega</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">ω</span></span></span></span></li> <li><code>\nabla</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">∇</mi></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">∇</span></span></span></span>, <code>\varphi</code>: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>φ</mi></mrow><annotation encoding="application/x-tex">\varphi</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit">φ</span></span></span></span></li></ul> <h2 id="latex常用大型公式模板"><a href="#latex常用大型公式模板" class="header-anchor">#</a> Latex常用大型公式模板</h2> <p>Latex条件公式显示如下:</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mrow><mo fence="true">{</mo><mtable><mtr><mtd><mrow><mn>1</mn><mo separator="true">,</mo><mspace width="1em"></mspace><msub><mi>d</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≤</mo><msubsup><mi>r</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>o</mi><mi>r</mi></mrow></msubsup></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn><mo separator="true">,</mo><mspace width="1em"></mspace><msub><mi>d</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>&gt;</mo><msubsup><mi>r</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>o</mi><mi>r</mi></mrow></msubsup></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">a_{ij}=
\begin{cases}
  1, \quad d_{ij} \le r^{coor}_i
  \\  
  0, \quad d_{ij} &gt; r^{coor}_i
\end{cases}
</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.75em;"></span><span class="strut bottom" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="minner displaystyle textstyle uncramped"><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist"><span style="top:-0.6819999999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord mathrm">1</span><span class="mpunct">,</span><span class="mord mspace quad"></span><span class="mord"><span class="mord mathit">d</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">≤</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="vlist"><span style="top:0.247em;margin-left:-0.02778em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">i</span></span></span><span style="top:-0.4129999999999999em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">c</span><span class="mord mathit">o</span><span class="mord mathit">o</span><span class="mord mathit" style="margin-right:0.02778em;">r</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:0.7579999999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord mathrm">0</span><span class="mpunct">,</span><span class="mord mspace quad"></span><span class="mord"><span class="mord mathit">d</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">&gt;</span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="vlist"><span style="top:0.247em;margin-left:-0.02778em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">i</span></span></span><span style="top:-0.4129999999999999em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">c</span><span class="mord mathit">o</span><span class="mord mathit">o</span><span class="mord mathit" style="margin-right:0.02778em;">r</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></span></p> <p>这种公式有两种写法，一种是使用<code>matrix</code>标签，另一种是使用<code>cases</code>标签，其中<code>cases</code>标签更加简洁。</p> <p><code>cases</code>标签的写法如下:</p> <div class="language-latex extra-class"><pre class="language-latex"><code><span class="token equation string">$$
a_{ij}=
<span class="token equation-command regex">\begin</span>{cases}
  1, <span class="token equation-command regex">\quad</span> d_{ij} <span class="token equation-command regex">\le</span> r^{coor}_i
  <span class="token equation-command regex">\\</span>  
  0, <span class="token equation-command regex">\quad</span> d_{ij} &gt; r^{coor}_i
<span class="token equation-command regex">\end</span>{cases}
$$</span>
</code></pre></div><p><code>matrix</code>标签的写法如下：</p> <div class="language-latex extra-class"><pre class="language-latex"><code><span class="token equation string">$$
a_{ij} = <span class="token equation-command regex">\left</span><span class="token equation-command regex">\{</span>
  <span class="token equation-command regex">\begin</span>{matrix}
    1, <span class="token equation-command regex">\quad</span> d_{ij} <span class="token equation-command regex">\le</span> r^{coor}_i
    <span class="token equation-command regex">\\</span>  
    0, <span class="token equation-command regex">\quad</span> d_{ij} &gt; r^{coor}_i
    <span class="token equation-command regex">\end</span>{matrix}<span class="token equation-command regex">\right</span>.
$$</span>
</code></pre></div><p>更复杂的多条件公式，实例1:</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtable><mtr><mtd><mrow><msup><mi>u</mi><mrow><mi>R</mi><mo>∗</mo></mrow></msup><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><mi>arg</mi><msub><mi>max</mi><mrow><msup><mi>u</mi><mi>R</mi></msup><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msub><msubsup><mi>J</mi><mi>r</mi><mi>R</mi></msubsup><mo fence="false">(</mo><msup><mi>x</mi><mi>R</mi></msup><mo>(</mo><mi>k</mi><mo>)</mo><mo separator="true">,</mo><msup><mi>u</mi><mi>R</mi></msup><mo>(</mo><mi>k</mi><mo>)</mo><mo fence="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mtext><mi mathvariant="normal">s</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">.</mi></mtext></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo fence="true">{</mo><mtable><mtr><mtd><mrow><msup><mi>x</mi><mi>R</mi></msup><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><msup><mi>x</mi><mi>R</mi></msup><mo>(</mo><mi>k</mi><mo>)</mo><mo separator="true">,</mo><msup><mi>u</mi><mi>R</mi></msup><mo>(</mo><mi>k</mi><mo>)</mo><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>N</mi><mi>R</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><msup><mi>G</mi><mi>R</mi></msup><mo fence="false">(</mo><msup><mi>x</mi><mi>R</mi></msup><mo>(</mo><mi>k</mi><mo>)</mo><mo separator="true">,</mo><msup><mi>u</mi><mi>R</mi></msup><mo>(</mo><mi>k</mi><mo>)</mo><mo fence="false">)</mo><mo>≤</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{array}{lcl}
  u^{R*}(k)=\arg\max\limits_{u^R(k)} J_r^R\big(x^R(k),u^R(k) \big)
  \\
  \text{s.t.}
  \\
  \begin{cases}
    x^R(k+1)=f(x^R(k), u^R(k))
    \\
    i=1,2,\ldots,N_R
    \\
    G^R\Big(x^R(k), u^R(k) \Big) \le 0
  \end{cases}
\end{array}
</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:4.188168em;"></span><span class="strut bottom" style="height:7.8763359999999984em;vertical-align:-3.688167999999999em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist"><span style="top:-3.296837em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">u</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mbin">∗</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mrel">=</span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mop op-limits"><span class="vlist"><span style="top:0.709965em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord"><span class="mord mathit">u</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span style="top:2.7755575615628914e-17em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="mop">max</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.09618em;">J</span><span class="vlist"><span style="top:0.247em;margin-left:-0.09618em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.02778em;">r</span></span></span><span style="top:-0.4129999999999999em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size1">(</span></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">u</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mord"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size1">)</span></span></span></span></span><span style="top:-1.4718720000000005em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="text mord displaystyle textstyle uncramped"><span class="mord mathrm">s</span><span class="mord mathrm">.</span><span class="mord mathrm">t</span><span class="mord mathrm">.</span></span></span></span><span style="top:1.5381479999999987em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="minner displaystyle textstyle uncramped"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing mult"><span class="vlist"><span style="top:1.25001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:1.25001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:0.95001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-0.000010000000000065512em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-1.15001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.45001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.7500200000000001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist"><span style="top:-1.5820099999999995em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mbin">+</span><span class="mord mathrm">1</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">u</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mclose">)</span></span></span><span style="top:-0.14200999999999964em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord mathit">i</span><span class="mrel">=</span><span class="mord mathrm">1</span><span class="mpunct">,</span><span class="mord mathrm">2</span><span class="mpunct">,</span><span class="minner">…</span><span class="mpunct">,</span><span class="mord"><span class="mord mathit" style="margin-right:0.10903em;">N</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.10903em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:1.4399899999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">G</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size2">(</span></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">u</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mord"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size2">)</span></span></span><span class="mrel">≤</span><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span></p> <p><strong>实例1</strong>代码如下：</p> <div class="language-latex extra-class"><pre class="language-latex"><code><span class="token equation string">$$
<span class="token equation-command regex">\begin</span>{array}{lcl}
  u^{R*}(k)=<span class="token equation-command regex">\arg</span><span class="token equation-command regex">\max</span><span class="token equation-command regex">\limits</span>_{u^R(k)} J_r^R<span class="token equation-command regex">\big</span>(x^R(k),u^R(k) <span class="token equation-command regex">\big</span>)
  <span class="token equation-command regex">\\</span>
  <span class="token equation-command regex">\text</span>{s.t.}
  <span class="token equation-command regex">\\</span>
  <span class="token equation-command regex">\begin</span>{cases}
    x^R(k+1)=f(x^R(k), u^R(k))
    <span class="token equation-command regex">\\</span>
    i=1,2,<span class="token equation-command regex">\ldots</span>,N_R
    <span class="token equation-command regex">\\</span>
    G^R<span class="token equation-command regex">\Big</span>(x^R(k), u^R(k) <span class="token equation-command regex">\Big</span>) <span class="token equation-command regex">\le</span> 0
  <span class="token equation-command regex">\end</span>{cases}
<span class="token equation-command regex">\end</span>{array}
$$</span>
</code></pre></div><p>实例2：</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>p</mi><mrow><mi>m</mi><mi>n</mi></mrow><mrow><mi>R</mi><mi>A</mi></mrow></msubsup><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable><mtr><mtd><mrow><mi>τ</mi><msubsup><mi>p</mi><mrow><mi>m</mi><mi>n</mi></mrow><mrow><mi>R</mi><mi>A</mi></mrow></msubsup><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mtd><mtd><mrow><mo>(</mo><mtext><mi mathvariant="normal">N</mi><mi mathvariant="normal">O</mi><mi mathvariant="normal">T</mi><mtext></mtext><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mtext><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><msubsup><mi>P</mi><mi>D</mi><mrow><mi>R</mi><mi>A</mi></mrow></msubsup><mo>⋅</mo><msubsup><mi>p</mi><mrow><mi>m</mi><mi>n</mi></mrow><mrow><mi>R</mi><mi>A</mi></mrow></msubsup><mo>(</mo><mi>k</mi><mo>)</mo></mrow><mrow><msubsup><mi>P</mi><mi>F</mi><mrow><mi>R</mi><mi>A</mi></mrow></msubsup><mo>+</mo><mo>(</mo><msubsup><mi>P</mi><mi>D</mi><mrow><mi>R</mi><mi>A</mi></mrow></msubsup><mo>−</mo><msubsup><mi>P</mi><mi>F</mi><mrow><mi>R</mi><mi>A</mi></mrow></msubsup><mo>)</mo><mo>⋅</mo><msubsup><mi>p</mi><mrow><mi>m</mi><mi>n</mi></mrow><mrow><mi>R</mi><mi>A</mi></mrow></msubsup><mi>i</mi><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mtext><mi mathvariant="normal">D</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi><mtext></mtext><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><mtext></mtext></mtext><mi>b</mi><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msubsup><mi>P</mi><mi>D</mi><mrow><mi>R</mi><mi>A</mi></mrow></msubsup><mo>)</mo><mo>⋅</mo><msubsup><mi>p</mi><mrow><mi>m</mi><mi>n</mi></mrow><mrow><mi>R</mi><mi>A</mi></mrow></msubsup><mo>(</mo><mi>k</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>−</mo><msubsup><mi>P</mi><mi>F</mi><mrow><mi>R</mi><mi>A</mi></mrow></msubsup><mo>+</mo><mo>(</mo><msubsup><mi>P</mi><mi>F</mi><mrow><mi>R</mi><mi>A</mi></mrow></msubsup><mo>−</mo><msubsup><mi>P</mi><mi>D</mi><mrow><mi>R</mi><mi>A</mi></mrow></msubsup><mo>)</mo><mo>⋅</mo><msubsup><mi>p</mi><mrow><mi>m</mi><mi>n</mi></mrow><mrow><mi>R</mi><mi>A</mi></mrow></msubsup><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mtext><mi mathvariant="normal">D</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi><mtext></mtext><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><mtext></mtext></mtext><mi>b</mi><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">p_{mn}^{RA}(k+1)=
\begin{cases}
  \tau p_{mn}^{RA}(k)  &amp; (\text{NOT detected})
  \\
  \frac{P_D^{RA} \cdot p_{mn}^{RA}(k)}{P_F^{RA} + (P_D^{RA} - P_F^{RA}) \cdot p_{mn}^{RA} i(k)} &amp; (\text{Detected and }b(k)=1)
  \\
  \frac{(1-P_D^{RA}) \cdot p_{mn}^{RA}(k)}{1 - P_F^{RA} + (P_F^{RA} - P_D^{RA}) \cdot p_{mn}^{RA}(k)} &amp; (\text{Detected and } b(k)=0)
\end{cases}
</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:3.494993em;"></span><span class="strut bottom" style="height:6.489986em;vertical-align:-2.9949929999999996em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">p</span><span class="vlist"><span style="top:0.247em;margin-left:0em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">m</span><span class="mord mathit">n</span></span></span></span><span style="top:-0.4129999999999999em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">A</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mbin">+</span><span class="mord mathrm">1</span><span class="mclose">)</span><span class="mrel">=</span><span class="minner displaystyle textstyle uncramped"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing mult"><span class="vlist"><span style="top:1.8500100000000002em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:1.8500100000000002em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:1.55001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:1.25001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:0.95001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-0.000010000000000065512em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-1.15001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.45001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.75001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.05001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.3500199999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist"><span style="top:-2.486993em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord mathit" style="margin-right:0.1132em;">τ</span><span class="mord"><span class="mord mathit">p</span><span class="vlist"><span style="top:0.247em;margin-left:0em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">m</span><span class="mord mathit">n</span></span></span></span><span style="top:-0.4129999999999999em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">A</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span><span style="top:-0.5366620000000002em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord reset-textstyle displaystyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.713131em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle cramped"><span class="mord textstyle cramped"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="vlist"><span style="top:0.29353099999999993em;margin-left:-0.13889em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.13889em;">F</span></span></span><span style="top:-0.34480000000000005em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">A</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="vlist"><span style="top:0.29353099999999993em;margin-left:-0.13889em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span></span></span><span style="top:-0.34480000000000005em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">A</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="vlist"><span style="top:0.29353099999999993em;margin-left:-0.13889em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.13889em;">F</span></span></span><span style="top:-0.34480000000000005em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">A</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mbin">⋅</span><span class="mord"><span class="mord mathit">p</span><span class="vlist"><span style="top:0.247em;margin-left:0em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">m</span><span class="mord mathit">n</span></span></span></span><span style="top:-0.28900000000000003em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">A</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">i</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="vlist"><span style="top:0.275331em;margin-left:-0.13889em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span></span></span><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">A</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">⋅</span><span class="mord"><span class="mord mathit">p</span><span class="vlist"><span style="top:0.247em;margin-left:0em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">m</span><span class="mord mathit">n</span></span></span></span><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">A</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span><span style="top:1.9883309999999996em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord reset-textstyle displaystyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.713131em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle cramped"><span class="mord textstyle cramped"><span class="mord mathrm">1</span><span class="mbin">−</span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="vlist"><span style="top:0.29353099999999993em;margin-left:-0.13889em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.13889em;">F</span></span></span><span style="top:-0.34480000000000005em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">A</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="vlist"><span style="top:0.29353099999999993em;margin-left:-0.13889em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.13889em;">F</span></span></span><span style="top:-0.34480000000000005em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">A</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="vlist"><span style="top:0.29353099999999993em;margin-left:-0.13889em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span></span></span><span style="top:-0.34480000000000005em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">A</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mbin">⋅</span><span class="mord"><span class="mord mathit">p</span><span class="vlist"><span style="top:0.247em;margin-left:0em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">m</span><span class="mord mathit">n</span></span></span></span><span style="top:-0.28900000000000003em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">A</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped"><span class="mord textstyle uncramped"><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mbin">−</span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="vlist"><span style="top:0.275331em;margin-left:-0.13889em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span></span></span><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">A</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mbin">⋅</span><span class="mord"><span class="mord mathit">p</span><span class="vlist"><span style="top:0.247em;margin-left:0em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">m</span><span class="mord mathit">n</span></span></span></span><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.00773em;">R</span><span class="mord mathit">A</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist"><span style="top:-2.486993em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mopen">(</span><span class="text mord displaystyle textstyle uncramped"><span class="mord mathrm">N</span><span class="mord mathrm">O</span><span class="mord mathrm">T</span><span class="mord mspace"></span><span class="mord mathrm">d</span><span class="mord mathrm">e</span><span class="mord mathrm">t</span><span class="mord mathrm">e</span><span class="mord mathrm">c</span><span class="mord mathrm">t</span><span class="mord mathrm">e</span><span class="mord mathrm">d</span></span><span class="mclose">)</span></span></span><span style="top:-0.5366620000000002em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mopen">(</span><span class="text mord displaystyle textstyle uncramped"><span class="mord mathrm">D</span><span class="mord mathrm">e</span><span class="mord mathrm">t</span><span class="mord mathrm">e</span><span class="mord mathrm">c</span><span class="mord mathrm">t</span><span class="mord mathrm">e</span><span class="mord mathrm">d</span><span class="mord mspace"></span><span class="mord mathrm">a</span><span class="mord mathrm">n</span><span class="mord mathrm">d</span><span class="mord mspace"></span></span><span class="mord mathit">b</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span><span style="top:1.9883309999999996em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mopen">(</span><span class="text mord displaystyle textstyle uncramped"><span class="mord mathrm">D</span><span class="mord mathrm">e</span><span class="mord mathrm">t</span><span class="mord mathrm">e</span><span class="mord mathrm">c</span><span class="mord mathrm">t</span><span class="mord mathrm">e</span><span class="mord mathrm">d</span><span class="mord mspace"></span><span class="mord mathrm">a</span><span class="mord mathrm">n</span><span class="mord mathrm">d</span><span class="mord mspace"></span></span><span class="mord mathit">b</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathrm">0</span><span class="mclose">)</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></span></p> <p><strong>实例2</strong>代码如下：</p> <div class="language-latex extra-class"><pre class="language-latex"><code><span class="token equation string">$$
p_{mn}^{RA}(k+1)=
<span class="token equation-command regex">\begin</span>{cases}
  <span class="token equation-command regex">\tau</span> p_{mn}^{RA}(k)  &amp; (<span class="token equation-command regex">\text</span>{NOT detected})
  <span class="token equation-command regex">\\</span>
  <span class="token equation-command regex">\frac</span>{P_D^{RA} <span class="token equation-command regex">\cdot</span> p_{mn}^{RA}(k)}{P_F^{RA} + (P_D^{RA} - P_F^{RA}) <span class="token equation-command regex">\cdot</span> p_{mn}^{RA} i(k)} &amp; (<span class="token equation-command regex">\text</span>{Detected and}b(k)=1)
  <span class="token equation-command regex">\\</span>
  <span class="token equation-command regex">\frac</span>{(1-P_D^{RA}) <span class="token equation-command regex">\cdot</span> p_{mn}^{RA}(k)}{1 - P_F^{RA} + (P_F^{RA} - P_D^{RA}) <span class="token equation-command regex">\cdot</span> p_{mn}^{RA}(k)} &amp; (<span class="token equation-command regex">\text</span>{Detected and} b(k)=0)
<span class="token equation-command regex">\end</span>{cases}
$$</span>
</code></pre></div><p>大型矩阵显示如下：</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>A</mi><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msub><mo>=</mo><mrow><mo fence="true">[</mo><mtable><mtr><mtd><mrow><msub><mi>a</mi><mrow><mn>1</mn><mn>1</mn></mrow></msub></mrow></mtd><mtd><mrow><msub><mi>a</mi><mrow><mn>1</mn><mn>2</mn></mrow></msub></mrow></mtd><mtd><mrow><mo>⋯</mo></mrow></mtd><mtd><mrow><msub><mi>a</mi><mrow><mn>1</mn><mi>n</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>a</mi><mrow><mn>2</mn><mn>1</mn></mrow></msub></mrow></mtd><mtd><mrow><msub><mi>a</mi><mrow><mn>2</mn><mn>2</mn></mrow></msub></mrow></mtd><mtd><mrow><mo>⋯</mo></mrow></mtd><mtd><mrow><msub><mi>a</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mi mathvariant="normal">⋮</mi></mrow></mtd><mtd><mrow><mi mathvariant="normal">⋮</mi></mrow></mtd><mtd><mrow><mo>⋱</mo></mrow></mtd><mtd><mrow><mi mathvariant="normal">⋮</mi></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>a</mi><mrow><mi>m</mi><mn>1</mn></mrow></msub></mrow></mtd><mtd><mrow><msub><mi>a</mi><mrow><mi>m</mi><mn>2</mn></mrow></msub></mrow></mtd><mtd><mrow><mo>⋯</mo></mrow></mtd><mtd><mrow><msub><mi>a</mi><mrow><mi>m</mi><mi>n</mi></mrow></msub></mrow></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">A_{m\times n}=  
\begin{bmatrix}  
  a_{11} &amp; a_{12} &amp; \cdots  &amp; a_{1n} \\  
  a_{21} &amp; a_{22} &amp; \cdots  &amp; a_{2n} \\  
  \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\  
  a_{m1} &amp; a_{m2} &amp; \cdots  &amp; a_{mn}  
\end{bmatrix}  
=\left [ a_{ij}\right ]
</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:2.68em;"></span><span class="strut bottom" style="height:4.859999999999999em;vertical-align:-2.1799999999999997em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">A</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">m</span><span class="mbin">×</span><span class="mord mathit">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="minner displaystyle textstyle uncramped"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing mult"><span class="vlist"><span style="top:1.50501em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:0.35001000000000015em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.2509899999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.49701em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist"><span style="top:-1.84em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">1</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:-0.6399999999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:0.6199999999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord mathrm">⋮</span></span></span><span style="top:1.8199999999999996em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">m</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist"><span style="top:-1.84em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">1</span><span class="mord mathrm">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:-0.6399999999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span><span class="mord mathrm">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:0.6199999999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord mathrm">⋮</span></span></span><span style="top:1.8199999999999996em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">m</span><span class="mord mathrm">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist"><span style="top:-1.84em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="minner">⋯</span></span></span><span style="top:-0.6399999999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="minner">⋯</span></span></span><span style="top:0.6199999999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="minner">⋱</span></span></span><span style="top:1.8199999999999996em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="minner">⋯</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist"><span style="top:-1.84em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">1</span><span class="mord mathit">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:-0.6399999999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span><span class="mord mathit">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:0.6199999999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord mathrm">⋮</span></span></span><span style="top:1.8199999999999996em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">m</span><span class="mord mathit">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing mult"><span class="vlist"><span style="top:1.50501em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:0.35001000000000015em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.2509899999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.49701em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span class="mrel">=</span><span class="minner displaystyle textstyle uncramped"><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;">[</span><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="style-wrap reset-textstyle textstyle uncramped" style="top:0em;">]</span></span></span></span></span></span></p> <p>大型矩阵的写法如下所示：</p> <div class="language-latex extra-class"><pre class="language-latex"><code><span class="token equation string">$$
A_{m<span class="token equation-command regex">\times</span> n}=  
<span class="token equation-command regex">\begin</span>{bmatrix}  
  a_{11} &amp; a_{12} &amp; <span class="token equation-command regex">\cdots</span>  &amp; a_{1n} <span class="token equation-command regex">\\</span>  
  a_{21} &amp; a_{22} &amp; <span class="token equation-command regex">\cdots</span>  &amp; a_{2n} <span class="token equation-command regex">\\</span>  
  <span class="token equation-command regex">\vdots</span> &amp; <span class="token equation-command regex">\vdots</span> &amp; <span class="token equation-command regex">\ddots</span> &amp; <span class="token equation-command regex">\vdots</span> <span class="token equation-command regex">\\</span>  
  a_{m1} &amp; a_{m2} &amp; <span class="token equation-command regex">\cdots</span>  &amp; a_{mn}  
<span class="token equation-command regex">\end</span>{bmatrix}  
=<span class="token equation-command regex">\left</span> [ a_{ij}<span class="token equation-command regex">\right</span> ] 
$$</span>
</code></pre></div><p>多行等式一般显示如下:</p> <p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtable><mtr><mtd><mrow><msup><mn>3</mn><mrow><mn>6</mn><mi>n</mi><mo>+</mo><mn>3</mn></mrow></msup><mo>+</mo><msup><mn>4</mn><mrow><mn>6</mn><mi>n</mi><mo>+</mo><mn>3</mn></mrow></msup></mrow></mtd><mtd><mrow><mrow></mrow><mo>=</mo><mo>(</mo><msup><mn>3</mn><mn>3</mn></msup><msup><mo>)</mo><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>(</mo><msup><mn>4</mn><mn>3</mn></msup><msup><mo>)</mo><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mrow><mrow></mrow><mo>≡</mo><mn>2</mn><msup><mn>7</mn><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>6</mn><msup><mn>4</mn><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mrow><mrow></mrow><mo>≡</mo><mn>2</mn><msup><mn>7</mn><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>(</mo><mo>−</mo><mn>2</mn><mn>7</mn><msup><mo>)</mo><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mrow><mrow></mrow><mo>≡</mo><mn>2</mn><msup><mn>7</mn><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>2</mn><msup><mn>7</mn><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{aligned}
  3^{6n+3}+4^{6n+3}
  &amp; = (3^3)^{2n+1}+(4^3)^{2n+1} \\
  &amp; \equiv 27^{2n+1}+64^{2n+1} \\  
  &amp; \equiv 27^{2n+1}+(-27)^{2n+1} \\
  &amp; \equiv 27^{2n+1}-27^{2n+1} \\
\end{aligned}
</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:3.298216em;"></span><span class="strut bottom" style="height:6.096432em;vertical-align:-2.7982159999999996em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist"><span style="top:-2.434108em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord"><span class="mord mathrm">3</span><span class="vlist"><span style="top:-0.41300000000000003em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">6</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathrm">4</span><span class="vlist"><span style="top:-0.41300000000000003em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">6</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">3</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:-1.2099999999999997em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"></span></span><span style="top:0.01410799999999962em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"></span></span><span style="top:1.238215999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"></span></span><span style="top:2.4382159999999997em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="col-align-l"><span class="vlist"><span style="top:-2.434108em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord displaystyle textstyle uncramped"></span><span class="mrel">=</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">3</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.41300000000000003em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">2</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">4</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.41300000000000003em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">2</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:-1.2099999999999997em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord displaystyle textstyle uncramped"></span><span class="mrel">≡</span><span class="mord mathrm">2</span><span class="mord"><span class="mord mathrm">7</span><span class="vlist"><span style="top:-0.41300000000000003em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">2</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathrm">6</span><span class="mord"><span class="mord mathrm">4</span><span class="vlist"><span style="top:-0.41300000000000003em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">2</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:0.01410799999999962em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord displaystyle textstyle uncramped"></span><span class="mrel">≡</span><span class="mord mathrm">2</span><span class="mord"><span class="mord mathrm">7</span><span class="vlist"><span style="top:-0.41300000000000003em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">2</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathrm">2</span><span class="mord mathrm">7</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.41300000000000003em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">2</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:1.238215999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord displaystyle textstyle uncramped"></span><span class="mrel">≡</span><span class="mord mathrm">2</span><span class="mord"><span class="mord mathrm">7</span><span class="vlist"><span style="top:-0.41300000000000003em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">2</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord mathrm">2</span><span class="mord"><span class="mord mathrm">7</span><span class="vlist"><span style="top:-0.41300000000000003em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">2</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span></span></span></p> <p>代码如下：</p> <div class="language-latex extra-class"><pre class="language-latex"><code><span class="token equation string">$$
<span class="token equation-command regex">\begin</span>{aligned}
  3^{6n+3}+4^{6n+3}
  &amp; = (3^3)^{2n+1}+(4^3)^{2n+1} <span class="token equation-command regex">\\</span>
  &amp; <span class="token equation-command regex">\equiv</span> 27^{2n+1}+64^{2n+1} <span class="token equation-command regex">\\</span>  
  &amp; <span class="token equation-command regex">\equiv</span> 27^{2n+1}+(-27)^{2n+1} <span class="token equation-command regex">\\</span>
  &amp; <span class="token equation-command regex">\equiv</span> 27^{2n+1}-27^{2n+1} <span class="token equation-command regex">\\</span>
<span class="token equation-command regex">\end</span>{aligned}
$$</span>
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